Estimation theory

In any experimental platform, data acquisition is the first and essential step, and occasionally the most time-consuming and costly operation. During the process of data acquisition, we conduct experiments to measure the response of the system to a set of inputs. Methods of optimal design of experiment can be used to determine the most informative measurements and avoid numerous traps that trial-and-error experimentation might cause. In this research, we have developed a general approach for designing optimal experiments, subsequently applying it to the domain of optical tomography.
Optical tomography is a vital technology that enables three-dimensional imaging by reconstructing images from two-dimensional projections. This technology has applications across various fields, including medicine and material science. The process involves two main phases: data acquisition and image reconstruction. The traditional raster scanning method has been the standard approach for data acquisition, but it presents challenges in terms of scanning speed, quality, and exposure to harmful radiations in some cases. This has prompted researchers to explore ways to optimize the scanning process.

Determining the variance of a statistic (such as the sample median) can be
difficult. Various methods of Bootstrapping (re-sampling with replacement) were
used to estimate variance of one or more statistics based on a single sample. This
estimator was compared to the empirical estimators based on repeated simulations of
various sample sizes from a given distribution. Of particular interest was which of the
methods of Bootstrapping were most effective with a dependent data set. Different
degrees of dependency were used for the simulations with dependent data.

In recent years more and more researchers have begun to use data mining and
machine learning tools to analyze gene microarray data. In this thesis we have collected a
selection of datasets revolving around prediction of patient response in the specific area
of breast cancer treatment. The datasets collected in this paper are all obtained from gene
chips, which have become the industry standard in measurement of gene expression. In
this thesis we will discuss the methods and procedures used in the studies to analyze the
datasets and their effects on treatment prediction with a particular interest in the selection
of genes for predicting patient response. We will also analyze the datasets on our own in
a uniform manner to determine the validity of these datasets in terms of learning potential
and provide strategies for future work which explore how to best identify gene signatures.

It is known that response probability densities, although important in failure analysis, are seldom achievable for stochastically excited systems except for linear systems under additive excitations of Gaussian processes. Most often, statistical moments are obtainable analytically or experimentally. It is proposed in this thesis to determine the probability density from the known statistical moments using artificial neural networks. A multi-layered feed-forward type of neural networks with error back-propagation training algorithm is proposed for the purpose and the parametric method is adopted for identifying the probability density function. Three examples are given to illustrate the applicability of the approach. All three examples show that the neural network approach gives quite accurate results in comparison with either the exact or simulation ones.

This thesis demonstrates the theory and the application of
the Kalman filter model, a model where the coefficients are
not assumed to be constant over time but time-varying. This
model is approached in two different ways. The first is the
recursive approach and the second is the Mixed estimation
approach. Both of these two approaches describe in
different ways the original Kalman filter model. This thesis
also contains an empirical application of the Kalman filter
model, using real data from the Greek economy to estimate
the Demand for Money.

This thesis demonstrates the theory and logic of the ridge
regression method of statistical estimation. The effect of
multicollinearity on ordinary least squares is explored,
and the optimality of the ridge estimators is derived. The
critical analyses of ridge regression that have been
developed in the biased estimation literature and by
Bayesian statisticians is discussed. The technique of
ridge regression is compared to ordinary least squares in
the context of estimating price and income elasticities for
Greek imports.